Games on Strange Boards

Tic-Tac-Toe, Checkers, Chess

Many common games are played on a square or rectangular board with a grid on it. This board can be thought of as part of the Euclidean plane of classical geometry. We would like to come up with new boards to play these games on which are still ``locally Euclidean'' -- they should still be made up of a grid, but they might have large-scale features which make them different from the usual boards.

Cylindrical Tic-Tac-Toe

Play a few games of Tic-Tac-Toe on a cylinder. You can still play on the usual board by imagining the right edge joined to the left edge. Compared to normal Tic-Tac-Toe, there are four new ways to win.

What are the four new ways to win?

What is the best strategy for each player?

Toroidal Tic-Tac-Toe

If we also join the top edge to the bottom edge, we get a shape called a torus.

Try to visualize this shape.

Are there new ways to win toroidal Tic-Tac-Toe compared to standard Tic-Tac-Toe? How about compared to cylindrical Tic-Tac-Toe?

How would a game of chess work out on a torus if we started from the usual position?

Can you come up with better starting positions for toroidal chess?

Tic-Tac-Toe on Other Surfaces

There are many other ways of identifying edges of the board that lead to interesting shapes. Play some games of Tic-Tac-Toe on these boards or, for a bigger challenge, try checkers or chess on them. On some of these boards, a chess queen or rook could get behind a line of defense by moving sideways. Which boards have this property?

Mobius Strip

Klein Bottle

Projective Plane

3D Tic-Tac-Toe

You can also play Tic-Tac-Toe on a cube. What is the best strategy on this board? A harder game can be played by trying for 4-in-a-row on a 4x4x4 cube. Once you have played a few games of 3D Tic-Tac-Toe, try playing on the 3-torus; this is the space you get by identifying opposite sides of the cube, just as the torus is obtained by identifying opposite sides of the square.

Surfaces of Higher Genus

Roughly speaking, the genus of a surface is the number of donut-like holes in it. A sphere has genus zero, while the torus has genus 1. By gluing edges of polygons with more than 4 sides, we can create surfaces of higher genus. For example, gluing opposite edges of an octagon creates a surface of genus two. Can you think of games to play on the octagon? Try playing these games on the genus two surface as well.